National Course Specification

Course Assessment details have been updated to reflect

changes to the Course Assessment.

National Unit Specification

Unit: Statistics (Higher)

D325 12 This Unit has been removed from these Arrangements as Course C102 12 Mathematics: Maths 1, 2 and Stats deleted as this Course has been withdrawn. Please note that this Unit has been retained and can be offered as a freestanding Unit.

C100 12 Mathematics: Maths 1, 2 and 3

This course consists of three mandatory units as follows:

D321 12 Mathematics 1 (H) 1 credit (40 hours)

In common with all courses, this course includes 40 hours over and above the 120 hours for thecomponent units. This may be used for induction, extending the range of learning and teachingapproaches, support, consolidation, integration of learning and preparation for external assessment.This time is an important element of the course and advice on its use is included in the course details.

RECOMMENDED ENTRYWhile entry is at the discretion of the centre, candidates will normally be expected to have attainedone of the following:• Standard Grade Mathematics Credit award• Intermediate 2 Mathematics or its component units including Mathematics 3 (Int 2)• equivalent.

*SCQF points are used to allocate credit to qualifications in the Scottish Credit and QualificationsFramework (SCQF). Each qualification is allocated a number of SCQF credit points at an SCQFlevel. There are 12 SCQF levels, ranging from Access 1 to Doctorates.

CORE SKILLS

This course gives automatic certification of the following:

Complete core skills for the course Numeracy H

Additional core skills components for the course Critical Thinking H

For information about the automatic certification of core skills for any individual unit in this course,please refer to the general information section at the beginning of the unit.

Mathematics: Higher Course 3

RATIONALEAs with all mathematics courses, Higher Mathematics aims to build upon and extend candidates’mathematical skills, knowledge and understanding in a way that recognises problem solving as anessential skill and enables them to integrate their knowledge of different aspects of the subject.

Because of the importance of these features, the grade descriptions for mathematics emphasise theneed for candidates to undertake extended thinking and decision making to solve problems andintegrate mathematical knowledge. The use of coursework tasks to achieve the course gradedescriptions in problem solving is encouraged.

The increasing degree of importance of mathematical rigour and the ability to use precise and concisemathematical language as candidates progress in mathematics assumes a particular importance at thisstage. Candidates who complete a Higher Mathematics course successfully are expected to have acompetence and a confidence in applying mathematical techniques, manipulating symbolicexpressions and communicating with mathematical correctness in the solution of problems. It isimportant, therefore, that, within the course, appropriate attention is given to the acquisition of suchexpertise whilst extending the candidate’s ‘toolkit’ of knowledge and skills.

Where appropriate, mathematics should be developed in context and the use of mathematicaltechniques should be applied in social and vocational contexts related to likely progression routes.

The Higher Mathematics course has the particular objective of meeting the needs of candidates at astage of their education where career aspirations are particularly important. The course has obviousrelevance for candidates with interests in fields such as commerce, engineering and science where themathematics learned will be put to direct use. For other candidates, the course can be used to gainentry to a Higher Education institution. All candidates taking the Higher Mathematics course,whatever their career aspirations, should acquire an enhanced awareness of the importance ofmathematics to technology and to society in general.

COURSE CONTENTThe syllabus is designed to build upon prior learning in the areas of algebra, geometry andtrigonometry, and to introduce candidates to elementary calculus.Mathematics 1 (H), Mathematics 2 (H) and Mathematics 3 (H) are progressive units, each unitcovering the range of mathematical topics described above.

The outcomes and the performance criteria for each unit are statements of basic competence.Additionally, the course makes demands over and above the requirements of individual units.Candidates should be able to integrate their knowledge across the component units of the course.Some of the 40 hours of flexibility time should be used to ensure that candidates satisfy the gradedescriptions for mathematics courses which involve solving problems and which require moreextended thinking and decision making. Candidates should be exposed to coursework tasks whichrequire them to interpret problems, select appropriate strategies, come to conclusions andcommunicate intelligibly.

Mathematics: Higher Course 4

Where appropriate, mathematical topics should be taught and skills in applying mathematicsdeveloped through real-life contexts. Candidates should be encouraged, throughout the course, tomake efficient use of the arithmetical, mathematical and graphical facilities on calculators. At thesame time, candidates should also be aware of the limitations of the technology and of the importanceof always applying the strategy of checking.

Numerical checking or checking a result against the context in which it is set is an integral part ofevery mathematical process. In many instances, the checking can be done mentally, but on occasions,to stress its importance, there should be evidence of a checking procedure throughout themathematical process. There are various checking procedures which could be used:• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order.

It is expected that candidates will be able to demonstrate attainment in the algebraic, trigonometricand calculus content of the course without the use of computer software or sophisticated calculators.

In assessments, candidates are required to show their working in carrying out algorithms andprocesses.

Mathematics: Higher Course 5

National Course Specification: course details (cont)DETAILED CONTENTThe content listed below should be covered in teaching the course. All of this content will be subject to sampling in the external assessment. Part of thisassessment will be carried out in a question paper where a calculator will not be allowed. Any of the topics may be sampled in this part of the assessment.The external assessment will also assess problem solving skills, see the grade descriptions on pages 25 and 26. Where comment is offered, this is intended tohelp in the effective teaching of the course.

References shown in this style of print indicate the depth of treatment appropriate to Grades A and B.

CONTENT COMMENT

Symbols, terms and sets

the symbols: ∈, ∉, { }

the terms: set, subset, empty set, member, element

the conventions for representing sets, namely:

N, the set of natural numbers, {1, 2, 3, ...}W, the set of whole numbers, {0, 1, 2, 3, ...}Z, the set of integersQ, the set of rational numbersR, the set of real numbers

The content listed above is not examinable but candidates are expected to be ableto understand its use.

Mathematics 1 (H)Properties of the straight lineknow that the gradient of the line determined by the points (x1, y1) and (x2, y2) isy2 − y1 , x2 ≠ x1 and that the distance between the points is ( x2 − x1 ) 2 + ( y2 − y1 ) 2x2 − x1

Mathematics: Higher Course 6

National Course Specification: course details (cont)

CONTENT COMMENTknow that the gradient of a straight line is the tangent of the angle made bythe line and the positive direction of the x-axis (scales being equal)

recognise the term locus

know that the equation of a straight line is of the form ax + by + c = 0, and

conversely (a, b not both zero)

know that the line through (x1, y1) with gradient m has equationy – y1 = m(x – x1)

determine the equation of a straight line given two points on the line or onepoint and the gradient

know that the gradients of parallel lines are equal

know that the lines with non-zero gradients m1 and m2 are perpendicular if andonly if m1m2 = –1

know the concurrency properties of medians, altitudes, bisectors of the angles Candidates should meet all of these concurrency facts, but they could beand perpendicular bisectors of the sides of a triangle covered by particular numerical cases only, using plane coordinate methods. However, they can also be established by vector methods and it is possible to explore this work through an investigative approach using an interactiveFunctions and graphs geometry package.know the meaning of the terms: domain and range of a function, inverse of afunction and composite function The main use of ‘inverse’ is to obtain the logarithmic function from theknow the meaning of the terms amplitude and period exponential function. An extended treatment is not required.

be aware of the general features of the graphs of f : x → sin (ax + b),

x → cos (ax + b) for suitable constants a, b

Mathematics: Higher Course 7

National Course Specification: course details (cont)

CONTENT COMMENTh given the graph of f(x) draw the graphs of related functions, where f(x) is a Candidates should be fluent in associating functions with their graphs and simple polynomial or trigonometric function vice versa: simple polynomial functions and trigonometric functions.

eg y = – f(x), f(x) + 2, 3f(x), f(x + 3), f(x – 1), f ′ ( x ).

π eg y = 3f(x) + 2, f(3x + 2), –3 f(x), 3f(x + ). [A/B] 2

know the general features of the graphs of the functions: Candidates should be fluent in associating functions with their graphs and vice versa: logarithmic and exponential functions. f: x → ax (a > 1 and 0 < a < 1, x ∈ R) Candidates should recognise that (12) x = 2-x. f: x → logax (a > 1, x > 0) Graphs of y = ax for a = 2, 3, etc., and for a = 1 2 , 1 3, etc. should be explored, and the inverse function of f: x → ax (a > 1, x ∈ R) should be investigated.

Inverse should be used to obtain the logarithmic function from the

exponential function.

use the notation f(g(x)) for a composite function and find composite functions Note: notation f o g is not required. of the form f(g(x)), given f(x) and g(x)

recognise the probable form of a function from its graph Candidates should be able to interpret a diagram containing the sketches of two (or possibly more) relationships, and sketch and annotate the graphs of functions once the critical features, such as points where the graph cuts the coordinate axes and axes of symmetry, as appropriate, have been identified.

Mathematics: Higher Course 8

National Course Specification: course details (cont)

CONTENT COMMENT 2complete the square in a quadratic of the form x + px + q The completed form would be (x + a)2 + b where a is an integer.complete the square in a quadratic of the form ax2 + bx + c [A/B] eg 2x2 – x + 1. [A/B]

Mathematics: Higher Course 9

National Course Specification: course details (cont)

CONTENT COMMENT

Basic differentiationknow the meaning of the terms limit, differentiable at a point, differentiate, Introduce within a context, such as investigating gradients of tangents atderivative, differentiable over an interval, derived function points on a curve. Gradients of a sequence of chords could be found numerically using a calculator. Opportunities for a cooperative approach dy should be taken.use the notation f ′(x ) and for a derivative dx

f ( x + h) − f ( x )know that f ′(x ) = lim h →0 h This could be deferred until rules have been consolidated in examples and applications. Note: differentiation from first principles is not included.

know the meaning of the terms rate of change, average gradient, strictly Rates of change (equations of motion, for example) are often expressed withincreasing, strictly decreasing, stationary point (value), maximum turning respect to time. Work should include other rates, eg rate of change of volumepoint (value), minimum turning point (value), horizontal point of inflexion of a sphere with respect to radius.

Mathematics: Higher Course 10

National Course Specification: course details (cont)

CONTENT COMMENT

know that f ′(a ) is the rate of change of f at a

know that f ′(a ) is the gradient of the tangent to the curve y = f (x) at Many relationships in science are expressed in terms of rates of change.x=a Candidates should be able to translate from words to symbols, and vice versa.

know that the gradient of the curve y = f(x) at any point on the curve is the It is important that the recognised convention for scientific symbols isgradient of the tangent at that point followed to avoid confusion when candidates meet symbols in other subject areas, eg use s (in lower case) for distance and v for speed and velocity.find the gradient of the tangent to a curve y = f(x) at x = a

find the points on a curve at which the gradient has a given value

know and apply the fact that:

if f ′( x ) > 0 in a given interval then the function f is strictly increasing in that interval

if f ′( x ) < 0 in a given interval then the function f is strictly decreasing

in that interval

if f ′(a ) = 0, then the function f has a stationary value at x = a

find the stationary point(s) (values) on a curve and determine their nature The most common method would be a nature table involving f ′.using differentiation

sketch a curve with given equation by finding stationary point(s) and theirnature, intersections with the axes, behaviour of y for large positive andnegative values of x

Mathematics: Higher Course 11

National Course Specification: course details (cont)

CONTENT COMMENT

determine the greatest/least values of a function on a given interval eg Given y = f(x) on an interval, differentiate and find values of end points, hence deduce maximum value.solve optimisation problems using calculus

Recurrence relationsknow the meaning of the terms: sequence, nth term, limit as n tends to Candidates should experience a variety of mathematical models of situationsinfinity involving recurrence relations.

use the notation un for the nth term of a sequence ‘Series’ is not included, but could be approached using sequence of partial sums.define and interpret a recurrence relation of the form un + 1 = mun + c( m, c constants) in a mathematical model

know the condition for the limit of the sequence resulting from a recurrencerelation to exist

find (where possible) and interpret the limit of the sequence resulting from arecurrence relation in a mathematical model

Mathematics: Higher Course 12

National Course Specification: course details (cont)

CONTENT COMMENTMathematics 2 (H)

Factor/Remainder Theorem and quadratic theory

use the Remainder Theorem to determine the remainder on dividing a ie Remainder is f(h).polynomial f (x) by x – h

determine the roots of a polynomial equation

use the Factor Theorem to determine the factors of a polynomial ie If f(h) = 0 then x – h is a factor of the polynomial f(x) and conversely.

Polynomial f(x) is of the third or higher degree, where f(x) can be expressed as a product of factors of which at most one is quadratic and the remainder are linear. eg f(x) = (x – 2)(x – 3)(2x + 1)(x – 1). ie no need to consider h ∈ Q for factor x – h. eg f(x) = (2x – 1)(3x + 2)(2x – 5). [A/B] − b ± b 2 − 4acknow that the roots of ax2 + bx + c = 0 , a ≠ 0, are 2aknow that the discriminant of ax2 + bx + c is b2 – 4ac

use the discriminant to:

determine whether or not the roots of a quadratic equation are real, and, if real, whether equal or unequal, rational or irrational This condition is to be applied in algebraic contexts when the equation is find the condition that the roots of a quadratic equation are real, and, if presented in a standard form, and in familiar geometric contexts. real, whether equal or unequal

Mathematics: Higher Course 13

National Course Specification: course details (cont)

CONTENT COMMENT

eg For what values of p does the equation x2 – 2x + p = 0 have real roots?

know the condition for tangency, intersection of a straight line and a The properties of the discriminant will also be required for the intersection ofparabola straight lines and circles, tangents to circles.

determine a quadratic equation with given roots The formulae for the product and the sum of the roots are not included.

prove that an equation has a root between two given values and find that rootto a required degree of accuracy

Mathematics: Higher Course 14

National Course Specification: course details (cont)

CONTENT COMMENT

Basic integrationknow the meaning of the terms integral, integrate, constant of integration, Introduce in a context such as investigating areas beneath curves. Calculatorsdefinite integral, limits of integration, indefinite integral, area under a curve could be used to determine approximations to areas. Conjectures made about general results should be investigated. Computer packages are available toknow that if f(x) = F ′(x) then ∫ f(x)dx = F(b) − F(a) and ∫ f(x) dx = F(x) + C assist with this approach. b

apply trigonometric formulae in the solution of geometric problems Applications include solution of triangles and three-dimensional situations, such as calculating the size of the angle between a line and a plane or between two planes. Work should extend to problems involving compound angles.

solve trigonometric equations involving addition formulae and double angle Candidates should be given the opportunity to find the general solution toformulae trigonometric equations, although in assessments solutions would be on a given interval, eg 0 ≤ θ ≤ 2π.

determine the equation of a circle

solve mathematical problems involving the intersection of a straight line and ie Determine the points at which a given line intersects a given circle,a circle, a tangent to a circle determine whether a given line is a tangent to a given circle, eg Show that the line with equation y = 2x – 10 is a tangent to the circle x2 + y2 – 4x + 2y = 0 and state the coordinates of the point of contact.

determine whether two circles touch each other eg The line with equation x – 3y = k is a tangent to the circle x2 + y2 – 6x + 8y + 15 = 0. Find the two possible values of k. [A/B]

Mathematics: Higher Course 17

National Course Specification: course details (cont)

CONTENT COMMENTMathematics 3 (H)

Vectors in three dimensions Suitable contexts for vectors could be forces and velocity.know the terms: vector, magnitude (length), direction, scalar multiple, positionvector, unit vector, directed line segment, component, scalar product Triangular concurrency facts can be established by vector methods. Some candidates should be encouraged to explore this work through anknow the properties of vector addition and multiplication of a vector by a scalar investigative approach, which would lead to simple vector proofs.

determine the distance between two points in three dimensional space

⎛ a⎞ ⎛ d ⎞ ⎜ ⎟ ⎜ ⎟know and apply the equality fact ⎜ b ⎟ = ⎜ e ⎟ ⇒ a = d, b = e, c = f * An understanding of both two and three dimensional vectors is expected. ⎜ ⎟ ⎜ ⎟ The items marked by an asterisk * are quoted for three dimensions but the ⎝c⎠ ⎝ f ⎠ two dimensional cases are also included.know and apply the fact that if u and v are vectors that can be represented by eg Determine the coordinates or position vector of the point which dividesparallel lines then u = kv where k is a constant and the converse the join of two given points in a given ratio.

AP mknow and apply the fact that if A, P and B are collinear points such that = The section formula may be used to find the position vector of P but is not PB n required. → m →then AP = PB n

determine whether three points with given coordinates are collinear

know and apply the basis vectors i, j, k * Understanding of the concepts should be reinforced by concentrating on numerical examples.

Mathematics: Higher Course 18

National Course Specification: course details (cont)

CONTENT COMMENT

know the scalar product facts:

determine whether or not two vectors, in component form, are perpendicular eg If |a| , |b| ≠ 0 then a . b = 0 if and only if the directions of a and b are at right angles.use scalar product to find the angle between two directed line segments

Mathematics: Higher Course 19

National Course Specification: course details (cont)

CONTENT COMMENT

Logarithmic and exponential functions

know that ay = x ⇔ logax = y (a > 1, x > 0)

know the laws of logarithms: Change of base is not included and when base is understood in a particular loga1 = 0 context, log x can be used for logax. The base a will normally be 10 or e and logaa = 1 the notation ln x should be introduced to candidates. loga(bc) = logab + logac b Numerical uses of logarithms and exponential evaluations are best done by loga( c ) = logab – logac calculators. loga(bn) = nlogab

solve for a and b equations of the following forms, given two pairs ofcorresponding values of x and y: log y = alog x + b, y = axb, y = abx[A/B]

use a straight line graph to confirm a relationship of the form

y = axb, also y = abx [A/B]

model mathematically situations involving the logarithmic or eg From experimental data draw a graph of log y against log x andexponential function [A/B] deduce values of a and b such that y = axb. [A/B]

Mathematics: Higher Course 20

National Course Specification: course details (cont)

CONTENT COMMENT

Further trigonometric relationships

express a cos θ + b sin θ in the form r cos(θ ± α) or r sin(θ ± α) Candidates should be encouraged to show all intermediate working, eg the calculation of r and α, the expansion of trigonometric formulae and thesolve, by expressing in one of the forms above, equations of the form equating of coefficients.a cos θ + b sin θ = c

find maximum and minimum values of expressions of the form

a cos θ + b sin θ find corresponding values of θ [A/B]

Mathematics: Higher Course 21

ASSESSMENTTo gain the award of the course, the candidate must pass all the unit assessments as well as theexternal assessment. External assessment will provide the basis for grading attainment in the courseaward.

When units are taken as component parts of a course, candidates should have the opportunity toachieve at levels beyond that required to attain each of the unit outcomes. This attainment may,where appropriate, be recorded and used to contribute towards course estimates and to provideevidence for appeals.

COURSE ASSESSMENTThe external assessment will take the form of an examination of 2 hours and 40 minutes duration. Theexternal examination will test the candidate’s ability to retain and integrate mathematical knowledgeacross the component units of the course. The examination will consist of two papers. Furtherinformation is available in the Course Assessment Specification and in the Specimen Question Paper.

Mathematics: Higher Course 22

National Course Specification: course details (cont)

COURSE Mathematics (Higher)

GRADE DESCRIPTIONSHigher Mathematics courses should enable candidates to solve problems that integrate mathematicalknowledge across performance criteria, outcomes and units, and which require extended thinking anddecision making. The award of Grades A, B and C is determined by the candidate’s demonstration ofthe ability to apply knowledge and understanding to problem-solving. To achieve Grades A and B inparticular, this demonstration will involve more complex contexts including the depth of treatmentindicated in the detailed content tables.

In solving problems, candidates should be able to:

a) interpret the problem and consider what might be relevant;b) decide how to proceed by selecting an appropriate strategy;c) implement the strategy through applying mathematical knowledge and understanding and come to a conclusion;d) decide on the most appropriate way of communicating the solution to the problem in an intelligible form.

Familiarity and complexity affect the level of difficulty of problems. It is generally easier to interpretand communicate information in contexts where the relevant variables are obvious and where theirinter relationships are known. It is usually more straightforward to apply a known strategy than tomodify one or devise a new one. Some concepts are harder to grasp and some techniques moredifficult to apply, particularly if they have to be used in combination.

Mathematics: Higher Course 23

Exemplification of problem solving at Grade C and Grade A

At Grade C candidates should be able to interpret qualitative and quantitative information as it

arises within: • the description of real-life situations • the context of other subjects • the context of familiar areas of mathematics.

Grade A performance is demonstrated through coping with the interpretation of more complex contexts requiring a higher degree of reasoning ability in the areas described above.

b) Decide how to proceed by selecting an appropriate strategy

At Grade C candidates should be able to tackle problems by selecting an algorithm, or sequence of algorithms, drawn from related areas of mathematics, or a heuristic strategy.

Grade A performance is demonstrated through an ability to decide on and apply a more extended sequence of algorithms to more complex contexts.

c) Implement the strategy through applying mathematical knowledge and understanding and come to a conclusion At Grade C candidates should be able to use their knowledge and understanding to carry through their chosen strategy and come to a conclusion. They should be able to process data in numerical and symbolic form with appropriate regard for accuracy, marshal facts, sustain logical reasoning and appreciate the requirements of proof.

Grade A performance is demonstrated through an ability to cope with processing data in more complex situations, and sustaining logical reasoning, where the situation is less readily identifiable with a standard form.

d) Decide on the most appropriate way of communicating the solution to the problem in an intelligible form At Grade C candidates should be able to communicate qualitative and quantitative mathematical information intelligibly and to express the solution in language appropriate to the situation.

Grade A performance is demonstrated through an ability to communicate intelligibly in more

complex situations and unfamiliar contexts.

Mathematics: Higher Course 24

APPROACHES TO LEARNING AND TEACHING

The learning and teaching process should foster positive attitudes to the subject. Exposition to agroup or class remains an essential technique at this level, and active candidate involvement inlearning should be encouraged through questioning and discussion. However, investigativeapproaches to learning should also feature prominently in the delivery of the Higher course. Whereappropriate, new skills and concepts should be introduced within a context and, when suitable,through an investigative approach, sometimes giving candidates the opportunity to work co-operatively. Coursework tasks can support these approaches, and simultaneously allow the gradedescriptions for extended problem solving to be met.

Opportunities should be taken to justify the need for new skills by reference to real problems. Whilethe contexts may include familiar and everyday ones, they should also make use of situations fromother subjects, or from areas of interest to candidates in their future study. Contexts which are purelymathematical will also be appropriate, for example, some aspects of quadratic theory might beillustrated in the context of tangency, Mathematics 2 (H).

Wherever possible, candidates should be encouraged to make use of technology. Calculators havegreat potential provided they are used appropriately and not allowed to provide unnecessary support,nor substitute for personal proficiency. This balance between ‘basic skills’ and appropriate use oftechnology is essential. Calculators with mathematical and graphical facilities and those withcomputer algebra systems (CAS) can be utilised as powerful tools both for processing data, especiallyin the study of statistics, and for reinforcing mathematical concepts. The use of such calculatorsshould help candidates gain confidence in making conjectures based on numerical or graphicalevidence. Candidates should be aware that errors are inevitably introduced in the course ofcomputation or in the limitations of the graphical display.

Computers can also make a significant contribution to learning and teaching. The use of softwarepackages in statistics will enhance the learning and teaching and allow candidates greater flexibilitythrough ease of computation and display.

It is envisaged that increased availability and advances in technology will have a continuing andincreasing influence in approaches to learning and teaching at this level.

CANDIDATES WITH DISABILITIES AND/OR ADDITIONAL SUPPORT NEEDS

The additional support needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments, or considering alternative Outcomes for Units.Further advice can be found in the SQA document Guidance on Assessment Arrangements forCandidates with Disabilities and/or Additional Support Needs (www.sqa.org.uk).

OUTCOMES1 Use the properties of the straight line.2 Associate functions and graphs.3 Use basic differentiation.4 Design and interpret mathematical models of situations involving recurrence relations.

RECOMMENDED ENTRYWhile entry is at the discretion of the centre, candidates will normally be expected to have attainedone of the following:• Standard Grade Mathematics Credit award• Intermediate 2 Mathematics or its component units including Mathematics 3 (Int 2)• equivalent.

Publication date: May 2007

Source: Scottish Qualifications Authority

This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived fromreproduction and that, if reproduced in part, the source is acknowledged.

Additional copies of this unit specification can be purchased from the Scottish Qualifications Authority. The cost foreach unit specification is £2.50 (minimum order £5). 26National Unit Specification: general information (cont)UNIT Mathematics 1 (Higher)

CREDIT VALUE

1 credit at Higher (6 SCQF credit points at SCQF level 6*).

*SCQF credit points are used to allocate credit to qualifications in the Scottish Credit andQualifications Framework (SCQF). Each qualification in the Framework is allocated a number ofSCQF credit points at an SCQF level. There are 12 SCQF levels, ranging from Access 1 toDoctorates.

Mathematics: Unit Specification - Mathematics 1 (H) 27

Acceptable performance in this unit will be the satisfactory achievement of the standards set out inthis part of the unit specification. All sections of the statement of standards are mandatory and cannotbe altered without reference to the Scottish Qualifications Authority.

OUTCOME 1Use the properties of the straight line.

Performance criteriaa) Determine the equation of a straight line given two points on the line or one point and the gradient.b) Find the gradient of a straight line using m = tan θ.c) Find the equation of a line parallel to and a line perpendicular to a given line.

OUTCOME 2Associate functions and graphs.

Performance criteriaa) Sketch and identify related graphs and functions.b) Identify exponential and logarithmic graphs.c) Find composite functions of the form f(g(x)), given f(x) and g(x).

OUTCOME 3Use basic differentiation.

Performance criteriaa) Differentiate a function reducible to a sum of powers of x.b) Determine the gradient of a tangent to a curve by differentiation.c) Determine the coordinates of the stationary points on a curve and justify their nature using differentiation.

Performance criteriaa) Define and interpret a recurrence relation in the form un+1 = mun + c (m, c constants) in a mathematical model.b) Find and interpret the limit of the sequence generated by a recurrence relation in a mathematical model (where the limit exists).

Evidence requirementsAlthough there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed book test under controlled conditions. Examples of suchtests are contained in the National Assessment Bank.

In assessment, candidates are required to show their working in carrying out algorithms andprocesses.

Mathematics: Unit Specification - Mathematics 1 (H) 28

National Unit Specification: support notesUNIT Mathematics 1 (Higher)

This part of the unit specification is offered as guidance. The support notes are not mandatory.

While the time allocated to this unit is at the discretion of the centre, the notional design length is40 hours.

GUIDANCE ON THE CONTENT AND CONTEXT FOR THIS UNIT

Each mathematics unit at Higher level aims to build upon and extend candidates’ mathematicalknowledge and skills. Within this unit, study of coordinate geometry of the straight line, algebra andtrigonometry, with the emphasis on graphicacy, is taken to a greater depth. Previous experience ofnumber patterns is formalised in the context of recurrence relations and differential calculus isintroduced.

The increasing degree of importance of mathematical rigour and the ability to use precise and concisemathematical language as candidates progress in mathematics assumes a particular importance at thisstage. Candidates working at this level are expected to acquire a competence and a confidence inapplying mathematical techniques, manipulating symbolic expressions and communicating withmathematical correctness in the solution of problems. It is important, therefore, that, within this unit,appropriate attention is given to the acquisition of such expertise whilst extending the candidate’s‘toolkit’ of knowledge and skills.

The recommended content for this unit can be found in the course specification. The detailed contentsection provides illustrative examples to indicate the depth of treatment required to achieve a unit passand advice on teaching approaches.

GUIDANCE ON LEARNING AND TEACHING APPROACHES FOR THIS UNIT

Where appropriate, mathematical topics should be taught and skills in applying mathematicsdeveloped through real-life contexts. Candidates should be encouraged throughout this unit to makeefficient use of the arithmetical, mathematical and graphical features of calculators, as well as basicnon-calculator skills. Candidates should be aware of the limitations of the technology and alwaysapply the strategy of checking.

Numerical checking or checking a result against the context in which it is set is an integral part ofevery mathematical process. In many instances, the checking can be done mentally, but on occasions,to stress its importance, attention should be drawn to relevant checking procedures throughout themathematical process. There are various checking procedures which could be used:• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order.

Further advice on learning and teaching approaches is contained within the Subject Guide forMathematics.

Mathematics: Unit Specification - Mathematics 1 (H) 29

GUIDANCE ON APPROACHES TO ASSESSMENT FOR THIS UNIT

The assessment for this unit will normally be in the form of a closed book test. Such tests should becarried out under supervision and it is recommended that candidates attempt an assessment designedto assess all the outcomes within the unit. Successful achievement of the unit is demonstrated bycandidates achieving the thresholds of attainment specified for all outcomes in the unit. Candidateswho fail to achieve the threshold(s) of attainment need only be retested on the outcome(s) where theoutcome threshold score has not been attained. Further advice on assessment and retesting iscontained within the National Assessment Bank.

It is expected that candidates will be able to achieve the algebraic, trigonometric and calculusperformance criteria of the unit without the use of computer software or sophisticated calculators.

In assessments, candidates should be required to show their working in carrying out algorithms andprocesses.

CANDIDATES WITH DISABILITIES AND/OR ADDITIONAL SUPPORT NEEDS

The additional support needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments, or considering alternative Outcomes for Units.Further advice can be found in the SQA document Guidance on Assessment Arrangements forCandidates with Disabilities and/or Additional Support Needs (www.sqa.org.uk).

Publication date: May 2007

Source: Scottish Qualifications Authority

This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived fromreproduction and that, if reproduced in part, the source is acknowledged.

Additional copies of this unit specification can be purchased from the Scottish Qualifications Authority. The cost foreach unit specification is £2.50 (minimum order £5). 31National Unit Specification: general information (cont)

UNIT Mathematics 2 (Higher)

CREDIT VALUE

1 credit at Higher (6 SCQF credit points at SCQF level 6*).

*SCQF credit points are used to allocate credit to qualifications in the Scottish Credit andQualifications Framework (SCQF). Each qualification in the Framework is allocated a number ofSCQF credit points at an SCQF level. There are 12 SCQF levels, ranging from Access 1 toDoctorates.

Core skills components for the unit Critical Thinking H

Mathematics: Unit Specification - Mathematics 2 (H) 32

National Unit Specification: statement of standards

UNIT Mathematics 2 (Higher)

Acceptable performance in this unit will be the satisfactory achievement of the standards set out inthis part of the unit specification. All sections of the statement of standards are mandatory and cannotbe altered without reference to the Scottish Qualifications Authority.

OUTCOME 1Use the Factor/Remainder Theorem and apply quadratic theory.

Performance criteriaa) Apply the Factor/Remainder Theorem to a polynomial function.b) Determine the nature of the roots of a quadratic equation using the discriminant.

OUTCOME 2Use basic integration.

Performance criteriaa) Integrate functions reducible to the sums of powers of x (definite and indefinite).b) Find the area between a curve and the x-axis using integration.c) Find the area between two curves using integration.

Performance criteriaa) Solve a trigonometric equation in a given interval.b) Apply a trigonometric formula (addition formula) in the solution of a geometric problem.c) Solve a trigonometric equation involving an addition formula in a given interval.

OUTCOME 4Use the equation of the circle.

Performance criteriaa) Given the centre (a, b) and radius r, find the equation of the circle in the form (x-a)² + (y-b)² = r².b) Find the radius and centre of a circle given the equation in the form x2 + y2 + 2gx + 2fy + c = 0.c) Determine whether a given line is a tangent to a given circle.d) Determine the equation of the tangent to a given circle given the point of contact.

Evidence requirementsAlthough there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed book test under controlled conditions. Examples of suchtests are contained in the National Assessment Bank.

In assessments, candidates are required to show their working in carrying out algorithms andprocesses.

Mathematics: Unit Specification - Mathematics 2 (H) 33

National Unit Specification: support notesUNIT Mathematics 2 (Higher)

This part of the unit specification is offered as guidance. The support notes are not mandatory.

While the time allocated to this unit is at the discretion of the centre, the notional design length is40 hours.

GUIDANCE ON THE CONTENT AND CONTEXT FOR THIS UNIT

Each mathematics unit at Higher level aims to build upon and extend candidates’ mathematicalknowledge and skills. Within this unit, the coordinate geometry of Mathematics 1(H) is extended toinclude the circle. The Factor/Remainder Theorem and quadratic theory and addition formulae areadded to previous experience in algebra and trigonometry respectively. Basic integration is introducedto extend the calculus of Mathematics 1(H).

The increasing degree of importance of mathematical rigour and the ability to use precise and concisemathematical language as candidates progress in mathematics assumes a particular importance at thisstage. Candidates working at this level are expected to acquire a competence and a confidence inapplying mathematical techniques, manipulating symbolic expressions and communicating withmathematical correctness in the solution of problems. It is important, therefore, that, within this unit,appropriate attention is given to the acquisition of such expertise whilst extending the candidate’s‘toolkit’ of knowledge and skills.

The recommended content for this unit can be found in the course specification. The detailed contentsection provides illustrative examples to indicate the depth of treatment required to achieve a unit passand advice on teaching approaches.

GUIDANCE ON LEARNING AND TEACHING APPROACHES FOR THIS UNIT

Where appropriate, mathematical topics should be taught and skills in applying mathematicsdeveloped through real-life contexts. Candidates should be encouraged throughout this unit to makeefficient use of the arithmetical, mathematical and graphical features of calculators, as well as basicnon-calculator skills. Candidates should be aware of the limitations of the technology and alwaysapply the strategy of checking.

Numerical checking or checking a result against the context in which it is set is an integral part ofevery mathematical process. In many instances, the checking can be done mentally, but on occasions,to stress its importance, attention should be drawn to relevant checking procedures throughout themathematical process. There are various checking procedures which could be used:• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order.

Further advice on learning and teaching approaches is contained within the Subject Guide forMathematics.

Mathematics: Unit Specification - Mathematics 2 (H) 34

GUIDANCE ON APPROACHES TO ASSESSMENT FOR THIS UNIT

The assessment for this unit will normally be in the form of a closed book test. Such tests should becarried out under supervision and it is recommended that candidates attempt an assessment designedto assess all the outcomes within the unit. Successful achievement of the unit is demonstrated bycandidates achieving the thresholds of attainment specified for all outcomes in the unit. Candidateswho fail to achieve the threshold(s) of attainment need only be retested on the outcome(s) where theoutcome threshold score has not been attained. Further advice on assessment and retesting iscontained within the National Assessment Bank.

It is expected that candidates will be able to achieve the algebraic, trigonometric and calculusperformance criteria of the unit without the use of computer software or sophisticated calculators.

In assessments, candidates are required to show their working in carrying out algorithms andprocesses.

CANDIDATES WITH DISABILITIES AND/OR ADDITIONAL SUPPORT NEEDS

The additional support needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments, or considering alternative Outcomes for Units.Further advice can be found in the SQA document Guidance on Assessment Arrangements forCandidates with Disabilities and/or Additional Support Needs (www.sqa.org.uk).

Publication date: May 2007

Source: Scottish Qualifications Authority

This publication may be reproduced in whole or in part for educational purposes provided that no profit is derived fromreproduction and that, if reproduced in part, the source is acknowledged.

Additional copies of this unit specification can be purchased from the Scottish Qualifications Authority. The cost foreach unit specification is £2.50 (minimum order £5). 36National Unit Specification: general information (cont)UNIT Mathematics 3 (Higher)

CREDIT VALUE

1 credit at Higher (6 SCQF credit points at SCQF level 6*).

*SCQF credit points are used to allocate credit to qualifications in the Scottish Credit andQualifications Framework (SCQF). Each qualification in the Framework is allocated a number ofSCQF credit points at an SCQF level. There are 12 SCQF levels, ranging from Access 1 toDoctorates.

Mathematics: Unit Specification – Statistics (H) 37

Acceptable performance in this unit will be the satisfactory achievement of the standards set out inthis part of the unit specification. All sections of the statement of standards are mandatory and cannotbe altered without reference to the Scottish Qualifications Authority.

OUTCOME 1Use vectors in three dimensions.

Performance criteriaa) Determine whether three points with given coordinates are collinear.b) Determine the coordinates of the point which divides the join of two given points internally in a given numerical ratio.c) Use the scalar product.

Evidence requirementsAlthough there are various ways of demonstrating achievement of the outcomes, evidence wouldnormally be presented in the form of a closed book test under controlled conditions. Examples of suchtests are contained in the National Assessment Bank.

In assessments, candidates are required to show their working in carrying out algorithms andprocesses.

Mathematics: Unit Specification – Statistics (H) 38

National Unit Specification: support notesUNIT Mathematics 3 (Higher)

This part of the unit specification is offered as guidance. The support notes are not mandatory.

While the exact time allocated to this unit is at the discretion of the centre, the notional design length is40 hours.

GUIDANCE ON THE CONTENT AND CONTEXT FOR THIS UNIT

Each mathematics unit at Higher level aims to build upon and extend candidates’ mathematicalknowledge and skills. This optional unit is designed to continue the study of mathematics in the areasof algebra, geometry, trigonometry and calculus, and to provide a basis for the study of moreadvanced mathematics and applied mathematics.

Within this unit, the coordinate geometry of Mathematics 1(H) and Mathematics 2(H) is broadenedinto a three dimensional context with the introduction of vector notation and applications. Calculus isfurther extended to include differentiation and integration of the sine and cosine functions and therules for integrating and differentiating composite functions. The exponential and logarithmicfunctions introduced graphically in Mathematics 1(H) are used in applications, and furthertrigonometry is studied in the context of the wave function.

The increasing degree of importance of mathematical rigour and the ability to use precise and concisemathematical language as candidates progress in mathematics assumes a particular importance at thisstage. Candidates working at this level are expected to acquire a competence and a confidence inapplying mathematical techniques, manipulating symbolic expressions and communicating withmathematical correctness in the solution of problems. It is important, therefore, that, within this unit,appropriate attention is given to the acquisition of such expertise whilst extending the candidate’s‘toolkit’ of knowledge and skills.

The recommended content for this unit can be found in the course specification. The detailed contentsection provides illustrative examples to indicate the depth of treatment required to achieve a unit passand advice on teaching approaches.

GUIDANCE ON LEARNING AND TEACHING APPROACHES FOR THIS UNIT

Where appropriate, mathematical topics should be taught and skills in applying mathematicsdeveloped through real-life contexts. Candidates should be encouraged throughout this unit to makeefficient use of the arithmetical, mathematical and graphical features of calculators as well as basicnon-calculator skills. Candidates should be aware of the limitations of the technology and alwaysapply the strategy of checking.

Numerical checking or checking a result against the context in which it is set is an integral part ofevery mathematical process. In many instances, the checking can be done mentally, but on occasions,to stress its importance, attention should be drawn to relevant checking procedures throughout themathematical process. There are various checking procedures which could be used:• relating to a context – ‘How sensible is my answer?’• estimate followed by a repeated calculation• calculation in a different order.

Further advice on learning and teaching approaches is contained within the Subject Guide forMathematics.

Mathematics: Unit Specification – Statistics (H) 39

GUIDANCE ON APPROACHES TO ASSESSMENT FOR THIS UNIT

The assessment for this unit will normally be in the form of a closed book test. Such tests should becarried out under supervision and it is recommended that candidates attempt an assessment designedto assess all the outcomes within the unit. Successful achievement of the unit is demonstrated bycandidates achieving the thresholds of attainment specified for all outcomes in the unit. Candidateswho fail to achieve the threshold(s) of attainment need only be retested on the outcome(s) where theoutcome threshold score has not been attained. Further advice on assessment and retesting iscontained within the National Assessment Bank.

It is expected that candidates will be able to achieve the algebraic, trigonometric and calculusperformance criteria of the unit without the use of computer software or sophisticated calculators.

In assessments, candidates are required to show their working in carrying out these algorithms andprocesses.

CANDIDATES WITH DISABILITIES AND/OR ADDITIONAL SUPPORT NEEDS

The additional support needs of individual candidates should be taken into account when planninglearning experiences, selecting assessment instruments, or considering alternative Outcomes for Units.Further advice can be found in the SQA document Guidance on Assessment Arrangements forCandidates with Disabilities and/or Additional Support Needs (www.sqa.org.uk).